192 ON THE COLOURS OF THICK PLATES.
were these circumstances materially different in the case of the two diffractions above mentioned, the rings might be modified, or might even disappear altogether.
Let us consider first the case of a concave mirror when the luminous point and its image are in the same plane perpendicular to the axis. In this case, if we consider any point P on the dimmed surface, and any point M in the plane of the rings, the angle of diffraction for the ray diffracted at emergence will be L3PM*. For the ray diffracted at entrance, the angle of diffraction measured in air will be LPM3, that is to say, MJP is the course of a ray in air which by regular refraction into glass would be brought into the direction of the ray diffracted at P. If G be the intersection of the axis and the plane of the rings, G will be the centre of the system, and the middle point of both the lines LLS and MM^ and therefore LM^ will be equal and parallel to MLy Hence, on account of the smallness of the obliquities, the angles of diffraction LPMQ, LSPM are sensibly equal, and their planes sensibly coincident, but the deviations take place in opposite directions. But between the two diffractions the light undergoes reflexion; and since the mutual inclination of two rays is reversed by reflexion, we must conceive the direction of deviation reversed in the first diffraction, in order to compare the circumstances of the two diffractions. Allowing for this reversion, we see that not only are the angles of diffraction sensibly coincident, but the directions of deviation are the same.
Accordingly, the interference connected with diffraction, and the interference which gives rise to the colours of thick plates, take place independently of each other. For, let /, T denote-the vibrations at M due to two streams of light diffracted by any particle of dust P on entering the glass, and passing on opposite sides of P; let J, J' denote the vibrations due to two streams diffracted at emergence, and passing on the same sides of P as I, /', respectively ; and let / -f /' denote the resultant of 7 and I', and similarly in other cases. Let % be the difference of phase corresponding to the retardation R, and a) the difference of phase of/, /', and therefore also of J", J', on account of the similarity of
* In speaking of angles of diffraction, such as L3PM", I shall distinguish between LJPM and MPL^ using the former notation to denote that the deviation takes place from PL.A to PM, and the latter to denote that it takes place from PM to PI/.{.